A late dispute over the estate tax produced allegations of bad faith from all sides.
After hours of haggling, Biden headed for the Capitol to brief the Democratic rank and file.
Earlier, McConnell had agreed with Obama that an overall deal was near. In remarks on the Senate floor, he suggested Congress move quickly to pass tax legislation and "continue to work on finding smarter ways to cut spending" next year.
The White House and Democrats initially declined the offer, preferring to prevent the cuts from kicking in at the Pentagon and domestic agencies alike. A two-month compromise resulted.
Officials in both parties said the agreement would prevent tax increases at incomes below $400,000 for individuals and $450,000 for couples.
At higher levels, the rate would rise to a maximum of 39.6 percent from the current 35 percent. Capital gains and dividends in excess of those amounts would be taxed at 20 percent, up from 15 percent.
The deal also would also raise taxes on the portion of estates exceeding $5 million to 40 percent. At the insistence of Republicans, the $5 million threshold would rise each year with inflation.
Much or all of the revenue to be raised through higher taxes on the wealthy would help hold down the amount paid to the Internal Revenue Service by the middle class.
In addition to preventing higher rates for most, the agreement would retain existing breaks for families with children, for low-earning taxpayers and for those with a child in college. Also, the two sides agreed to prevent the alternative minimum tax from expanding to affect an estimated 28 million households for the first time in 2013, with an average increase of more than $3,000. The law originally was designed to make sure millionaires did not escape taxes, but inflation has gradually exposed more and more households with lower earnings to its impact.
The legislation leaves untouched a scheduled 2 percentage point increase in the payroll tax, ending a temporary reduction enacted two years ago to help revive the economy.
Officials said the White House had succeeded in gaining a one-year extension of long-term unemployment benefits about to expire on an estimated two million jobless.
The legislation would delay for one year a 27 percent cut in fees for doctors who treat Medicare patients.
Also included is a provision to prevent a threatened spike in milk prices after the first of the year.
Even as time was running out, partisan agendas were evident.
Obama used his appearance not only to chastise Congress, but also to lay down a marker for the next round of negotiations early in 2013, when Republicans intend to seek spending cuts in exchange for letting the Treasury to borrow above the current debt limit of $16.4 trillion.
"Now, if Republicans think that I will finish the job of deficit reduction through spending cuts alone — and you hear that sometimes coming from them ... then they've got another think coming. ... That's not how it's going to work at least as long as I'm president," he said.
"And I'm going to be president for the next four years, I think," he added.
Obama's remarks irritated some Republicans.
Sen. John McCain of Arizona they would "clearly antagonize members of the House."
Associated Press writers Julie Pace, Andrew Taylor, Alan Fram and Ben Feller contributed to this report.